Statistical multiple diffuse scattering from rough surfaces in RHEED - beyond the distorted-wave Born approximation

In reflection high-energy electron diffraction (RHEED) of growing surfaces in molecular beam epitaxy (MBE), diffuse scattering is generated by atom vibrations, point vacancies and growth islands (or surface roughness). Most of the existing RHEED theories have been developed under the first-order diffuse scattering approximation, and thus they are restricted for surfaces whose roughness is relatively low. In fact, crystal surfaces grown by MBE are usually rough; the change of surface coverage from 0 to 1 monolayer accounts for the observed RHEED oscillation. In this paper, a formal dynamical theory of RHEED has been developed to calculate the diffuse scattering produced by both atom vibrations and point vacancies at surfaces. The theory is aimed at recovering the multiple diffuse scattering that has been dropped by the distorted-wave Born approximation (DWBA). With the inclusion of a complex potential in the dynamical calculation, a rigorous proof is given to show that the high-order diffuse scattering terms are recovered in the calculation using the equation originally derived under the DWBA. This conclusion establishes the basis for expanding the RHEED theories developed under the first-order diffuse scattering to cases where the degree of surface roughness is high, allowing dynamical calculation of RHEED rocking curves for any growing surface. The statistical time and structure averages over the distorted crystal potential are evaluated analytically before numerical calculation. The dynamic form factor is calculated with consideration of anisotropic surface atom vibration and point vacancies at a growing surface.